Difference between revisions of "User:Tohline/Math/EQ Toroidal04"

Latest revision as of 21:39, 1 July 2018

	<math>~(\nu - \mu + 1)P^\mu_{\nu + 1} (z)</math>	<math>~=</math>	<math>~ (2\nu + 1)z P_\nu^\mu(z) - (\nu + \mu)P^\mu_{\nu-1}(z) </math>
Abramowitz & Stegun (1995), p. 334, eq. (8.5.3)

NOTE: <math>~Q_\nu^\mu</math>, as well as <math>~P_\nu^\mu</math>, satisfies this same recurrence relation.

@@ Line 3: / Line 3: @@
 <tr>
 <td align="right">
-[[Image:LSU_Key.png|25px|link=http://www.vistrails.org/index.php/User:Tohline/Appendix/Equation_templates#Special_Function_Relationships]]
+[[Image:LSU_Key.png|25px|link=http://www.vistrails.org/index.php/User:Tohline/Appendix/Equation_templates#Relationships_Between_Various_Associated_Legendre_Functions]]
 </td>
    <td align="right">
-<math>~(\nu - \mu + 1)P^\mu_{\vu + 1}</math>
+<math>~(\nu - \mu + 1)P^\mu_{\nu + 1} (z)</math>
    </td>
    <td align="center">
@@ Line 13: / Line 13: @@
    <td align="left">
 <math>~
-e^{i \mu \pi} ~ (2\pi)^{-\frac{1}{2}} (z^2-1)^{\mu/2} ~\Gamma(\mu + \tfrac{1}{2})~\biggl\{
+(2\nu + 1)z P_\nu^\mu(z) - (\nu + \mu)P^\mu_{\nu-1}(z)
-\int_0^\pi (z - \cos t)^{-\mu - \frac{1}{2}} \cos[(\nu + \tfrac{1}{2})t] ~dt
--\cos(\nu\pi) \int_0^\infty (z + \cosh t)^{-\mu - \frac{1}{2}} e^{-(\nu + \frac{1}{2})t} ~dt
-\biggr\}
 </math>
    </td>
@@ Line 30: / Line 27: @@
 <tr>
    <td align="right">
-NOTE:  Both <math>~P_\nu^\mu</math> and <math>~Q_\nu^\mu</math> satisfy this same recurrence relation.
+NOTE:  <math>~Q_\nu^\mu</math>, as well as <math>~P_\nu^\mu</math>, satisfies this same recurrence relation.
    </td>
 </tr>
 </table>